A well-known subclass of chordal graphs is formed by proper interval graphs. Due to their very special structural properties, several problems proved hard to solve for interval graphs can have better solutions for this subclass. In this paper, we address the recognition problem, proposing an update of one of the first existing linear algorithms. The outcome is a simple and efficient algorithm. In addition, we present a certifying algorithm for the recognition of proper interval graphs

A bipartite graph on \(2n\) vertices is called bipancyclic if it contains cycles of every length from \(4\) to \(2n\). In this paper we address the question: what is the minimum number of edges in a bipancyclic graph? We present a simple analysis of some small orders using chord patterns.

A vertex set \(U \subset V\) in a connected graph \(G = (V, E)\) is a cutset if \(G – U\) is disconnected. If no proper subset of \(U\) is also a cutset of \(G\), then \(U\) is a minimal cutset. An \(\mathcal{MVC}\)-partition \(\pi = \{V_1, V_2, \dots, V_k\}\) of the vertex set \(V(G)\) of a connected graph \(G\) is a partition of \(V(G)\) such that every \(V_i \in \pi\) is a minimal cutset of \(G\). For an \(\mathcal{MVC}\)-partition \(\pi\) of \(G\), the \(\pi\)-graph \(G_{\pi}\), of \(G\) has vertex set \(\pi\) such that \(V’,V” \in \pi\) are adjacent in \(G_{\pi}\), if and only if there exist \(v’ \in V’\) and \(v” \in V”\) such that \(v’v” \in E(G)\). Graphs that are a \(\pi\)- graphs of cycles are characterized. A homomorphic image \(H\) of a graph \(G\) can be obtained from a partition \(\mathcal{P} = {V_1, V_2,…, V_k}\) of \(V(G)\) into independent sets such that \(V(H) = {v_1,v_2,…, v_k}\), where \(v_i\) is adjacent to \(v_j\); if and only if some vertex of \(V_j\) is adjacent to some vertex of \(V_j\) in \(G\). By investigating graphs \(H\) that are homomorphic images of the Cartesian product \(H\Box K_2\), it is shown that for every nontrivial connected graph \(H\) and every integer \(r \geq 2\), there exists an \(r\)-regular graph \(G\) such that \(H\) is a homomorphic image of \(G\). It is also shown that every nontrivial tree \(T\) is a homomorphic image of \(T\Box K_2\) but that not all graphs \(H\) are homomorphic images of \(H\Box K_2\).

A Hamiltonian graph \(G\) is said to be \(\ell\)-path-Hamiltonian, where \(\ell\) is a positive integer less than or equal to the order of \(G\), if every path of order \(\ell\) in \(G\) is a subpath of some Hamiltonian cycle in \(G\). The Hamiltonian cycle extension number of \(G\) is the maximum positive integer \(L\) for which \(G\) is \(\ell\)-path-Hamiltonian for every integer \(\ell\) with \(1 \leq \ell \leq L\). Hamiltonian cycle extension numbers are determined for several well-known cubic Hamiltonian graphs. It is shown that if \(G\) is a cubic Hamiltonian graph with girth \(g\), where \(3 \leq g \leq 7\), then \(G\) is \(\ell\)-path-Hamiltonian only if \(1 \leq \ell \leq g\).

In a red-blue coloring of a graph \(G\), every edge of \(G\) is colored red or blue. For two graphs \(F\) and \(H\), the Ramsey number \(R(F, H)\) of \(F\) and \(H\) is the smallest positive integer \(n\) such that every red-blue coloring of the complete graph \(K_n\) of order \(n\) results in either a subgraph isomorphic to \(F\) all of whose edges are colored red or a subgraph isomorphic to \(H\) all of whose edges are colored blue. While the study of Ramsey numbers has been a popular area of research in graph theory, over the years a number of variations of Ramsey numbers have been introduced. We look at several of these, with special emphasis on some of those introduced more recently.

For a graph \(G\) of size \(m\), a graceful labeling of \(G\) is an injective function \(f : V(G) \to \{0, 1, \dots, m\}\) that gives rise to a bijective function \(f’ : E(G) \to \{1, 2, \dots, m\}\) defined by \(f'(uv) = |f(u) – f(v)|\). A graph \(G\) is graceful if \(G\) has a graceful labeling. Over the years, a number of variations of graceful labelings have been introduced, some of which have been described in terms of colorings. We look at several of these, with special emphasis on some of those introduced more recently.

For a connected graph \(G\) of order at least \(3\), let \(c : E(G) \to \{1, 2, \dots, k\}\) be an edge coloring of \(G\) where adjacent edges may be colored the same. Then \(c\) induces a vertex coloring \(c’\) of \(G\) by assigning to each vertex \(v\) of \(G\) the set of colors of the edges incident with \(v\). The edge coloring \(c\) is called a majestic \(k\)-edge coloring of \(G\) if the induced vertex coloring \(c’\) is a proper vertex coloring of \(G\). The minimum positive integer \(k\) for which a graph \(G\) has a majestic \(k\)-edge coloring is the majestic chromatic index of \(G\) and denoted by \(\chi_{m}^{‘} (G)\). For a graph \(G\) with \(\chi_{m}^{‘}(G) = k\), the minimum number of distinct vertex colors induced by a majestic \(k\)-edge coloring is called the majestic chromatic number of \(G\) and denoted by \(\psi(G)\). Thus, \(\psi(G)\) is at least as large as the chromatic number \(\chi(G)\) of a graph \(G\). Majestic chromatic indexes and numbers are determined for several well-known classes of graphs. Furthermore, relationships among the three chromatic parameters \(\chi_m(G)\), \(\psi(G)\), and \(\chi(G)\) of a graph \(G\) are investigated.

This paper investigates vertex colorings of graphs such that some rainbow subgraph \(R\) and some monochromatic subgraph \(M\) are forbidden. Previous work focused on the case that \(R = M\). Here we consider the more general case, especially the case that \(M = K_2\).

Let \(R\) be a commutative ring with identity. For any integer \(k > 1\), an element is a \(k\)-zero divisor if there are distinct \(k\) elements including the given one, such that the product of all is zero but the product of fewer than all is nonzero. Let \(Z(R,k)\) denote the set of the \(k\)-zero divisors of \(R\). In this paper we consider rings which are not \(k\)-integral domains (i.e. \(Z(R,k)\) is nontrivial) with finite \(Z(R,k)\). We show that a uniform \(n\) exists such that \(a^n = 0\) for all elements \(a\) of the nil-radical \(N\) and deduce that a ring \(R\) which is not a \(k\)-integral domain with more than \(k\) minimal prime ideals and whose nil-radical is finitely generated is finite, if \(Z(R,k)\) is finite.

Recording actual user interactions with a system is often useful for testing software applications. User-session based test suites that contain records of such interactions often find a complementary set of faults compared to test suites created by testers. This work utilizes such test suites and presents a new prioritization method that extends the existing combinatorial two-way inter-window prioritization by introducing weights on the distance between windows. We examine how a window distance between a pair of the parameter-value tuples influences the fault detection effectiveness. We evaluate several approaches used to calculate weights. Results show improvement over the original two-way inter-window prioritization technique, while the comparison of different weighting approaches reveals that a negative linear weighting calculation generally performs better in our experiments. The study demonstrates that the distance between windows in a pair is an important factor to consider in test suite prioritization, and that distinguishing windows by their order in a test case also improves the fault detection rate compared to using window labels that were utilized in previous methods. This work provides motivation for future work to develop general n-way combinatorial distance-based prioritization methods that take into account space and processing time requirements to address potential issues with large test suites.